TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-28

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Hurwitz's zero-free limit theorem

Statement

Let $Ω \subseteq C$ be a complex domain, let each $f_{n} : Ω \to C$ be holomorphic and nowhere zero, and suppose $f_{n} \to f$ locally uniformly on $Ω$ . Then either $f \equiv 0$ on $Ω$ , or $f$ is nowhere zero on $Ω$ .

Facts & Assumptions

Given: A complex domain $Ω$ , holomorphic nowhere-zero functions $f_{n}$ on $Ω$ , and locally uniform convergence $f_{n} \to f$ .

[L1]

Locally uniform limits of holomorphic functions are holomorphic (Locally uniform limits of holomorphic functions are holomorphic and their derivatives converge locally uniformly).

[L2]

Near an isolated zero of the limit, sufficiently late approximants have the same total zero multiplicity (Locally uniform convergence preserves the total multiplicity near an isolated zero).

Proof

technique · direct

1.1

By [L1], the limit function $f$ is holomorphic on $Ω$ . If $f \equiv 0$ , the first alternative of the statement holds.

givenL1

2.1

Assume $f \equiv̸ 0$ and suppose toward a contradiction that $f (a) = 0$ at some point $a \in Ω$ . Then $a$ is an isolated zero of $f$ , so [L2] gives a disc about $a$ on which every sufficiently large $f_{n}$ has at least one zero. That contradicts the hypothesis that every $f_{n}$ is nowhere zero.

step 1.1L2assume-contradischarge-contradiction

3.1

Therefore, in the nonzero branch of step 1.1, the function $f$ has no zeros on $Ω$ . This is the second alternative.

step 1.1step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

J. Lebl, Guide to Cultivating Complex Analysis, §5.4, Exercise 5.4.14 (standard reference, not scraped)
R. W. Howell and J. H. Mathews, Complex Analysis, §8.7 (standard reference, not scraped)